Finite Group Databasebeta

← #80338

Subgroups — #80338

Order 384.

434 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5716C211yesmaximal, characteristic, complemented
2#5548C211yesmaximal, characteristic, complemented
3C3×#459C211yesmaximal, characteristic, complemented
4#5158C212yesmaximal, complemented
5#5154C212yesmaximal, complemented
6#5072C211yesmaximal, characteristic, complemented
7#5019C211yesmaximal, characteristic, complemented
8#4841C211yesmaximal, characteristic, complemented
9#4835C211yesmaximal, characteristic, complemented
10#4781C212yesmaximal, complemented
11#4765C212yesmaximal, complemented
12#3214—31nomaximal, Sylow
13C2×#237C2212yes—
14C2×#235—21no—
15C2×C3×D8C2212yes—
16C3×#127C2211yescharacteristic
17C3×C4×D4C2211yescharacteristic
18#940C2211yescharacteristic
19#938—21no—
20#938—21no—
21#938C2211yescharacteristic
22#936C2212yes—
23D4×Dic3—21no—
24D4×Dic3—21no—
25C2×#214C2212yes—
26C2×#213C2212yes—
27C4×#67—21no—
28C2×#204C2212yes—
29#895C2211yescharacteristic
30#888C2211yescharacteristic
31#885—21no—
32#884—21no—
33C4×D12C2211yescharacteristic
34C42×D3C2211yescharacteristic
35#869C2211yescharacteristic
36C3×C4⋊C8C2211yescharacteristic
37C3×#102C2212yes—
38#832C2212yes—
39(C2×D3)⋊C8C2212yes—
40#816C2212yes—
41#815C2212yes—
42#810C2212yes—
43C4⋊(C3⋊C8)C2211yescharacteristic
44#532—31no—
45#532—31no—
46#496—32no—
47#489—32no—
48#461—31no—
49#461—31no—
50#459—31no—
51#459D311yescharacteristic, complemented
52#449—32no—
53#449—32no—
54#432—31no—
55C2×C3×D4—21no—
56C2×C3×D4—21no—
57C2×C3×D4C2312yes—
58C2×C3×D4C2311yescharacteristic
59C22×C3×C4—21no—
60C2×#67—21no—
61C2×#67—22no—
62C2×#67—21no—
63C2×#67—21no—
64C22×Dic3—22no—
65#237—42no—
66#237—42no—
67#235—44no—
68C2×D12—21no—
69C2×D12C2311yescharacteristic
70C2×C4×D3—21no—
71C2×C4×D3D411yescharacteristic
72C2×C4×D3D411yescharacteristic
73C2×C4×D3C2311yescharacteristic
74C2×Dic6—21no—
75C2×Dic6C2312yes—
76C3×D8—42no—
77C2×C3×C8C2312yes—
78C3×C4⋊C4C2311yescharacteristic
79C3×C22⋊C4—21no—
80C3×C42C2311yescharacteristic
81#217—21no—
82#217—22no—
83#217—22no—
84#214—42no—
85#213—42no—
86(C2×D3)⋊C4—21no—
87(C2×D3)⋊C4—21no—
88C4⋊Dic3—21no—
89C4⋊Dic3—21no—
90C4⋊Dic3C2311yescharacteristic
91Dic3⋊C4—21no—
92C4×Dic3D411yescharacteristic
93C4×Dic3D411yescharacteristic
94C4×Dic3C2311yescharacteristic
95C2×C3⋊C8C2312yes—
96#204—42no—
97C2×#41—32no—
98C2×#41—61no—
99C2×SD16—32no—
100C2×SD16—32no—
101C2×D8—32no—
102C2×D8C2×D312yes—
103C4⋊Q8—31no—
104C4⋊Q8—31no—
105#127—31no—
106#127C2×D311yescharacteristic
107#124—32no—
108#124—61no—
109#121—31no—
110#121—31no—
111#121—61no—
112#121—61no—
113#121—61no—
114C4×D4—31no—
115C4×D4—61no—
116C4×D4—61no—
117C4×D4—61no—
118C4×D4C2×D311yescharacteristic
119C2×C42—31no—
120C4⋊C8—31no—
121C4⋊C8C2×D311yescharacteristic
122#103—32no—
123#103—32no—
124#102—32no—
125#102C2×D312yes—
126C22⋊C8—32no—
127#97—32no—
128C23×C3—22no—
129C23×C3—21no—
130C22×D3—21no—
131C22×D3C2×D411yescharacteristic
132C3×D4—22no—
133C3×D4—21no—
134C3×D4—21no—
135C3×D4—22no—
136C3×D4—42no—
137C3×D4—42no—
138C2×C3×C4—21no—
139C2×C3×C4—42no—
140C2×C3×C4C2×D411yescharacteristic
141C2×C3×C4C2×D411yescharacteristic
142C2×C3×C4C2411yescharacteristic
143#67—42no—
144#67—42no—
145#67—42no—
146#67—44no—
147C2×Dic3—21no—
148C2×Dic3—22no—
149C2×Dic3—44no—
150C2×Dic3C2×D411yescharacteristic
151C2×Dic3C2×D411yescharacteristic
152C2×Dic3C2×D411yescharacteristic
153D12—21no—
154D12—21no—
155D12—42no—
156C4×D3—22no—
157C4×D3—21no—
158C4×D3—21no—
159C4×D3—41no—
160C4×D3—42no—
161Dic6—22no—
162Dic6—22no—
163Dic6—42no—
164C3×C8—22no—
165C3⋊C8—22no—
166#41—124no—
167#41—122no—
168#41—122no—
169C2×Q8—32no—
170C2×Q8—61no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—31no—
174C2×D4—61no—
175C2×D4—61no—
176C2×D4—61no—
177C2×D4—61no—
178C2×D4—62no—
179C2×D4C22×D312yes—
180C2×D4C22×D311yescharacteristic
181C22×C4—21no—
182C22×C4—31no—
183C22×C4—31no—
184C22×C4—31no—
185C22×C4—62no—
186C22×C4—61no—
187SD16—122no—
188SD16—122no—
189D8—42no—
190D8—122no—
191C2×C8—32no—
192C2×C8C22×D312yes—
193C4⋊C4—31no—
194C4⋊C4—61no—
195C4⋊C4—61no—
196C4⋊C4—61no—
197C4⋊C4C22×D311yescharacteristic
198C22⋊C4—21no—
199C22⋊C4—61no—
200C22⋊C4—61no—
201C22⋊C4—62no—
202C22⋊C4—62no—
203C22⋊C4—61no—
204C42—31no—
205C42—31no—
206C42—31no—
207C42C22×D311yescharacteristic
208C22×C3—42no—
209C22×C3—41no—
210C22×C3—41no—
211C22×C3—42no—
212C22×C3—41no—
213C22×C3—42no—
214C22×C3C22×D411yescharacteristic
215C2×D3—21no—
216C2×D3—21no—
217C2×D3—21no—
218C2×D3—41no—
219C2×D3—41no—
220C2×D3—41no—
221C3×C4—21no—
222C3×C4—41no—
223C3×C4AGL(1,ℤ/8)12yes—
224C3×C4H(5,𝔽2)11yescharacteristic
225C3×C4H(5,𝔽2)11yescharacteristic
226Dic3—21no—
227Dic3—21no—
228Dic3—21no—
229Dic3—42no—
230Dic3—41no—
231C23—21no—
232C23—22no—
233C23—31no—
234C23—61no—
235Q8—62no—
236Q8—62no—
237Q8—122no—
238D4—21no—
239D4—22no—
240D4—22no—
241D4—21no—
242D4—42no—
243D4—42no—
244D4—61no—
245D4—61no—
246D4—122no—
247D4—122no—
248D4—122no—
249D4—122no—
250D4—124no—
251C2×C4—21no—
252C2×C4—31no—
253C2×C4—31no—
254C2×C4—31no—
255C2×C4—42no—
256C2×C4—62no—
257C2×C4—61no—
258C2×C4—61no—
259C2×C4—61no—
260C2×C4—62no—
261C2×C4—121no—
262C2×C4—124no—
263C2×C4—122no—
264C2×C4D3×D411yescharacteristic
265C2×C4D3×D411yescharacteristic
266C2×C4C23×D311yescharacteristic
267C8—22no—
268C8—62no—
269C2×C3—82no—
270C2×C3—81no—
271C2×C3#53511yescharacteristic
272C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
273C2×C3#57511yescharacteristic
274D3—41no—
275D3—81no—
276C22—41no—
277C22—42no—
278C22—42no—
279C22—41no—
280C22—42no—
281C22—41no—
282C22—61no—
283C22—61no—
284C22—61no—
285C22—121no—
286C22—121no—
287C22—121no—
288C22C2×D3×D411yescharacteristic
289C4—21no—
290C4—41no—
291C4—61no—
292C4—61no—
293C4—61no—
294C4—121no—
295C4—122no—
296C4#91112yes—
297C4#100411yescharacteristic
298C4#100911yescharacteristic
299C3#321411yescharacteristic, complemented, Sylow
300C2—81no—
301C2—82no—
302C2—121no—
303C2—241no—
304C2#560711yescharacteristic
305C2C2×#91111yescharacteristic
306C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#459C2
socleC22×C3C22×D4
hypercenter#459D3
hypocenterC3#3214
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #5716C2semidirect
2. #5548C2semidirect
3. C3×#459C2semidirect
4. #5158C2semidirect
5. #5154C2semidirect
6. #5072C2semidirect
7. #5019C2semidirect
8. #4841C2semidirect
9. #4835C2semidirect
10. #4781C2semidirect
11. #4765C2semidirect
51. #459D3semidirect
299. C3#3214semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#57162
#55482
C3×#4592
#51582
#51542
#50722
#50192
#48412
#48352
#47812
#47652
#32143